Final answer:
The student's question involves finding the minimum amount of fiber-optic cable needed to connect all homes in a rural network, which is solved by finding the Minimum Spanning Tree (MST) for the given graph. However, without the graph data provided, the exact solution and cable layout cannot be determined.
Step-by-step explanation:
The question involves using mathematics, specifically graph theory, to find the minimum amount of fiber-optic cable needed to connect all homes in a rural area. This is essentially a problem that requires finding the Minimum Spanning Tree (MST) of a graph, where each node represents a home, and each edge represents the possible connection between homes with their respective lengths. The most common algorithms for finding the MST are Kruskal's or Prim's algorithm.
To provide an accurate answer, it is necessary to have the graph depicting the nodes and edges, which is not provided in the question. Without the graph, the minimum amount of fiber-optic cable cannot be accurately determined.
Once the MST is found using the proper algorithm, the length of the cable will be the sum of the weights (lengths of the cable segments) of the edges included in the MST. The layout of the cable would follow the network formed by the MST, ensuring that the cable is laid out in an efficient manner that connects all homes with the least amount of cable.